2015/08/26 by С. Е. Степанов, Sergey Stepanov, Stepanov, Sergey +2
Mathematics · Physics and Astronomy · #53C20 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C20
paper · pdf · doi:10.48550/arxiv.1508.06418
openalex publication_date 2015/08/26 · arxiv created 2016/06/14 · arxiv updated 2016/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fifty years ago, Eells and Sampson have proved a famous theorem in which they argued that any harmonic mapping f:(M,g) → (M,g) is totally geodesic if (M, g) is a compact manifold with the nonnegative Ricci tensor and the section curvature of (M,g) is nonpositive. Moreover, other main results of the theory of harmonic mappings "in the large" are the results on harmonic maps into nonpositively curved Riemannian manifolds. In our paper we develop a theory of harmonic mappings into Riemannian manifolds with nonnegative sectional curvature. In particular, we will prove that any harmonic map between Riemannian manifolds f:(M,g) → (M,g) is totally geodesic if the section curvature of (M,g) is nonnegative and (M, g) is a compact manifold with the Ricci tensor Ric ≥ f*Ric for the pullback f*Ric of the Ricci tensor Ric by f. The above scheme will be extended to a harmonic mapping of a complete manifold to a manifold with the nonnegative sectional curvature. Moreover, we will obtain interesting corollaries from our results.